Problem 89 - Algebra and Trigonometry with Analytic Geometry

Question

Use fundamental identities to find the values of the trigonometric functions for the given conditions. \(\cos \theta=-\frac{1}{3}\) and \(\sin \theta<0\)

Step-by-Step Solution

Answer

\( \sin \theta = -\frac{2\sqrt{2}}{3} \), \( \tan \theta = 2\sqrt{2} \).

Step 1: Determine the Quadrant

Given the conditions \( \cos \theta = -\frac{1}{3} \) and \( \sin \theta < 0 \), we know cosine is negative and sine is also negative. This places angle \( \theta \) in the third quadrant.

Step 2: Use Pythagorean Identity

We use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute the given value of cosine: \( \left(-\frac{1}{3}\right)^2 + \sin^2 \theta = 1 \). Simplify to find: \( \frac{1}{9} + \sin^2 \theta = 1 \).

Step 3: Solve for \( \sin \theta \)

Rearrange the equation to solve for \( \sin^2 \theta \): \( \sin^2 \theta = 1 - \frac{1}{9} \). This gives \( \sin^2 \theta = \frac{8}{9} \). Taking the square root, \( \sin \theta = -\frac{\sqrt{8}}{3} = -\frac{2\sqrt{2}}{3} \).

Step 4: Find \( \tan \theta \)

Using the definition \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), substitute \( \sin \theta = -\frac{2\sqrt{2}}{3} \) and \( \cos \theta = -\frac{1}{3} \) to get: \( \tan \theta = \frac{-\frac{2\sqrt{2}}{3}}{-\frac{1}{3}} = 2\sqrt{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The most commonly used trigonometric functions are sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). These functions are pivotal in understanding and solving problems involving angles and distances.

• Sine (\( \sin \theta \)): This function represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.


• Cosine (\( \cos \theta \)): It describes the ratio of the length of the adjacent side to the hypotenuse.


• Tangent (\( \tan \theta \)): This is the ratio of the sine of the angle to the cosine of the angle, or alternatively, the opposite side over the adjacent side.

Understanding how these ratios work is crucial for identifying the correct values in various quadrant positions on the unit circle. Depending on the quadrant, these functions can produce positive or negative values. Recognizing these sign changes is essential when determining the trigonometric values of angles based on given conditions.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry, highlighting the intrinsic connection between sine and cosine functions. This identity is expressed as:\[\sin^2 \theta + \cos^2 \theta = 1\]This equation illustrates that for any angle \( \theta \), the square of the sine plus the square of the cosine will always equal 1. It is derived from the Pythagorean theorem, hence its name.

• This identity can be utilized to find missing trigonometric values. For example, if you know \( \cos \theta \), you can determine \( \sin \theta \) and vice versa.


• In practical scenarios, given one trigonometric value, the Pythagorean identity is invaluable for computing other related functions.

In the exercise, knowing \( \cos \theta = -\frac{1}{3} \) enabled us to find \( \sin \theta \), reinforcing the power and utility of this identity in solving trigonometric problems.

Third Quadrant

The third quadrant is a sector in the Cartesian plane where both the x and y coordinates are negative. In terms of trigonometric functions, both the sine and cosine values are negative in this quadrant. This information is crucial for solving trigonometric problems where the angle lies in the third quadrant.

• In this quadrant, since both \( \sin \theta \) and \( \cos \theta \) are negative, the tangent function,\( \tan \theta = \frac{\sin \theta}{\cos \theta} \), becomes positive.


• Understanding this sign pattern is essential for correctly determining the resultant values of the functions \( \sin \theta \), \( \cos \theta \), and \( \tan \theta \) when given conditions that imply the third quadrant.

In our original exercise, identifying that the angle \( \theta \) fell in the third quadrant was a vital step. It provided the necessary information that both sine and cosine values are negative, ensuring the correct application of trigonometric functions and identities.